Question

How do you simplify `5(35(3^n-1) - 29(2^n-1) ) - 6(35(3^n-2) - 29(2^n-2) )`?

Answer 1

In the unlikely event that I didn't make an arithmetic mistake:
`color(white)("XXXX")-35(3^n)+275(2^n)+42`

Explanation:

Note: This isn't really difficult; it's just tedious.

`5(35(3^n−1)−29(2^n−1))−6(35(3^n−2)−29(2^n−2))`

`=(5xx35)(3^n-1)-(5xx29)(2^n-1) + ((-6)xx35)(3^n-2)-(-6xx29)(2^n-2)`

`=175(3^n-1)-145(2^n-1)+ (-210)(3^n-2) + 174(2^n-2)`

`=175*color(red)(3^n) color(green)(- 175)-145*color(blue)(2^n)color(green)(+145)-210*color(red)(3^n)color(green)(+420)+174*color(blue)(2^n)color(green)(-348)`

`=(175-210)*color(red)(3^n)+(-145+420)*color(blue)(2^n)+(color(green)(-175+145+420-348))`

`=-35(3^n)+275(2^n)+42`